1. Field of the Invention
The present invention is directed to an adaptive system for converging solutions, and more particularly, to a multidimensional adaptive system having relatively small dimensionality that can be made to converge to solutions that could otherwise only be converged by systems having much larger dimensionality.
2. Description of the Related Art
Multi-channel feedforward adaptive systems are, for example, used to cancel noise. However, certain factors affect convergence in adaptive systems. These include the step size parameter, generally designated as .mu., and the effectiveness of the filtering that must be inserted into the reference-signal path at the input to a weight-iteration stage to compensate for plant transfer functions between secondary sources and detection points for a filtered-X LMS algorithm. Compensation in a reference signal path in conventional systems must be identical to the forward transfer-function between the secondary sources and the detection points. When this occurs, the adaptive filter ideally converges to the Wiener solution. In addition, feedback between the secondary sources (actuators) and the reference-signal detectors is also a factor. However, these effects can be eliminated by neutralization and are not considered further.
A one-dimensional conventional system is shown in FIG. 1. In FIG. 1, error sensors (subtractors) 20 are provided which receive disturbance or target signals D to be cancelled or reduced, and a cancelling or error reduction signal produced by the system. The error sensors 20 then produce an error signal E=PWX-D. X is a reference signal, Q* is a compensation unit 22, .DELTA.W is an updating unit 24, W is an adaptive filter 26, and P is a physical plant 28 in which signals from the adaptive filter 26 must propagate before being input to the error sensors 20. P can vary with time. The reference signal X is input to the compensation unit 22 and the adaptive filter 26. The disturbance signals D are input to the error sensors 20. The error signals E from the error sensors 20 are input to the updating unit 24 along with compensated reference signals from the compensation unit 22. This combined signal is then input to the adaptive filter 26 along with the reference signal X and output to the physical plant 28. The physical plant 28 then outputs a signal PWX to the error sensors 20 which also receive the disturbance signals D. Thus, a feedback loop is established to compensate for the disturbance signals D, i.e., to cancel the disturbance signals.
Analysis of the one-dimensional system shown in FIG. 1, will now be given. The analysis will be carried out in frequency space with discrete Fourier transforms X(m) and D(m) of a discrete-time-series reference. Disturbance signals are given as x(n) and d(n) and W.sub.k (m) is the transfer function of the kth iteration of the adaptive-filter impulse response w.sub.k (n), given by ##EQU1## With the above formulation all results that follow are understood to refer to specific frequency pockets. Realistically, however, because of finite system bandwidth, finite ranges of frequencies should be considered.
Suppressing the discrete-frequency index m, the filter output W.sub.k X drives a secondary source L (not shown in FIG. 1) producing a response PW.sub.k X at the detection point, where the physical plant 28 generates a forward transfer function between the secondary source and the detection point which yields the squared error .vertline.D-PW.sub.k X.vertline..sup.2. In the conventional filtered-X LMS algorithm, where X is a reference signal and LMS is the least mean square, the transfer function from the physical plant 28 is compensated in the reference signal path prior to the updating unit 24 by P. The compensation operation is denoted by Q*. The weight-iteration equation for the filtered-X LMS algorithm in frequency space takes the form EQU W.sub.k+1 =W.sub.k +2.mu.Q*D-PW.sub.k X!X* (2)
and after applying the above expectation operator (eq.(2)) EQU W.sub.k+1 =W.sub.k +2.mu.Q*T-PW.sub.k S! (3)
where T=DX* is the cross spectral density between the reference and disturbance signals and S=XX* is the cross spectral density between the reference signals themselves.
The solution to the above difference equation (3) is EQU W.sub.k -W.sub.I !=W.sub.0 -W.sub.1 !1-2.mu.Q*P.vertline.X.vertline..sup.2 !.sup.k ( 4)
where W.sub.0 is the initial setting of the adaptive-filter transfer function at t=0 and W.sub.I =T/P.vertline.X.vertline..sup.2 is the ideal Wiener solution. With perfect compensation Q*=P* and the system converges if .vertline.1-2.mu..vertline.P.vertline..sup.2 .vertline.X.vertline..sup.2 .vertline.&lt;1, or equivalently EQU .mu..vertline.P.vertline..sup.2 .vertline.X.vertline..sup.2 &lt;1(5)
Now let EQU Q*P=.vertline.QP.vertline.e.sup.i.theta. =Ae.sup.i.theta.
If the phase mismatch is zero, the system still converges if EQU .mu.A.vertline.X.vertline..sup.2 &lt;1 (6)
On the other hand, in the presence of phase mismatch, the compensation equation (4) becomes EQU W.sub.k -W.sub.I !=W.sub.0 -W.sub.I !1-2.mu.Ae.sup.i.theta. .vertline.X.vertline..sup.2 !.sup.k= W.sub.0 -W.sub.I !1+4.mu..sup.2 A.sup.2 .vertline.X.vertline..sup.2 !.sup.2 -4.mu.A.vertline.X.vertline..sup.2 cos.theta.!.sup.k/2 e.sup.ik.phi.( 7)
where ##EQU2## If .vertline..theta..vertline.&gt;.pi./2, the magnitude of the term within the brackets in equation (7) exceeds unity and the system will not converge. Even if .vertline..theta..vertline.&lt;.pi./2, phase mismatch can be quite serious, and in this case, convergence EQU .mu.A.vertline.X.vertline..sup.2 &lt;cos.theta. (9)
requiring, compared with equation (6), possibly smaller values of .mu. to insure convergence. Also, even if the condition for convergence is met, the convergence time can be significantly increased since the term in square brackets in equation (7) increases with .theta.. The results are summarized in FIG. 2, including FIGS. 2A and 2B. As shown, if there is no phase mismatch the term 1-2.mu.Q.sup..dagger. P.vertline.X.vertline..sup.2 in equation (4) is real. Therefore, the phase of (W.sub.k -W.sub.0) remains unchanged during convergence and W.sub.k follows the shortest straight-line path from W.sub.0 to W.sub.I as shown in FIG. 2A. This is an essential feature of the filtered-X LMS algorithm. However, if there is phase mismatch the system may never converge and, at best, convergence will be slowed down as shown in FIG. 2B, with W.sub.k taking a circuitous route, as determined by .phi., through the complex plane from W.sub.0 to W.sub.I. Although mismatch in the compensation amplitude is tolerable, accurate phase compensation is critical.
Further, when a multidimensional system is employed, rather than a one-dimensional system as set forth above, the system can become very large to the point of becoming prohibitively large, expensive, less efficient and almost impossible to cancel noise.